IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. XX, NO. XX, XXX 2013 1 Model based compensation of thermal disturbance in a precision linear electromagnetic actuator Jonathan Hey, Choon Meng Kiew, Guilin Yang, Ricardo Martinez-Botas Abstract—Thermal disturbance is a major source of positioning error in precision positioning systems. Conventional approach of using special materials for construction and sophisticated geometric design based on advanced computer simulation can be costly as well as time consuming to implement. Moreover, dynamic thermal disturbances cannot be effectively compensated for by such methods. The approach presented in this paper uses model estimated position error coupled with sensor measurement as a feedback compensator of the output shaft position. A state space model is used in a Modified Kalman Filter (MKF) to reduce the total number of temperature sensors needed for estimation of the thermally induced position error. A maximum temperature estimation error of 1.8% of the measurement range is recorded. An online parameter estimation method is implemented to ‘fine tune’ a transfer function model during a calibration stage before compensation. The model based compensation method resulted in a mean unidirectional positioning deviation of -0.2μm and repeatability of ±0.7μm during a 5 hour continuous operation. Keywords—Compensation, Identification, Kalman Filter, Ther- moelastic I. I NTRODUCTION T HERMAL disturbance is a major source of position- ing error in precision positioning systems. Positioning error arises due to the internal or external thermal loading which leads to thermal strain. Material deformation due to temperature changes causes deviation of the output position from the desired position [1]. Such thermal error have been minimized by using construction material with low coefficient of thermal expansion (CTE) and crafted geometric designs to ensure the thermal center is at the output shaft [2]. Some have cooling system which actively maintains a constant operating temperature. However, such approaches can be costly, difficult to implement and not all the dynamic thermal error can be eliminated by these methods [1]. Fig. 1 shows a linear electromagnetic actuator designed for submicron positioning accuracy. Such actuators are common in precision stages which require long range travel but they are susceptible to disturbances [3]. However, the test device uses an electromagnetic driving scheme coupled with flexure- based supporting bearings to improve the position accuracy [4]. Nevertheless, it still suffers from thermal disturbance due to the Manuscript received xxxxx, 2013. This work is supported by the Agency for Science, Technology and Research (A*STAR) of Singapore. Jonathan Hey and Ricardo Martinez-Botas are with the Mechanical En- gineering Department, Imperial College London, Exhibition Road, London, SW7 2AZ, UK (e-mail: [email protected], [email protected]) Choon Meng Kiew and Guilin Yang are with Mechatronics group, Singapore Institute of Manufacturing Technology, 71 Nanyang Drive, Singapore, 638075 (e-mail: [email protected], [email protected]) internal heat generation at the coil during operation. Thermal expansion of the output shaft leads to eventual positioning error. A model based estimation of the disturbance coupled with active output compensation is a cost effective method to minimize the effects of dynamic disturbance [2],[5]. The relationship between temperature increase and the resulting thermal disturbance is determined from experimental data through model identification techniques [6],[7]. Fig. 1: Flexure based electromagnetic linear actuator Recent advances in this area of research focuses on system- atic reduction of temperature sensors required for modeling [8]. Parallel work on optimal temperature estimate of the rotor coil in an induction machine is shown using Kalman filtering technique [9]. Strategies for fine tuning model estimation ac- curacy is reported in [7], [10] which shows the trend of online parameter estimation using iterative algorithms. In this paper, a model based compensation method is presented which shows its effectiveness in minimizing thermal disturbance in the test device. The compensation model is experimentally determined from the measured input output data using the subspace and least square method for model parameter estimation. Internal temperature distribution is estimated using a state space model coupled with measurement feedback from an embedded temperature sensor in the coil. A Modified Kalman Filter (MKF) is presented to show how sensor reduction can be achieved in such an application. The thermo elastic process is modeled using this temperature information as input to a transfer function model. A recursive parameter estimation method is used to tune the transfer function model before compensation is carried out in an extended 5 hour test to simulate continuous operations. The purpose of this work is to show how the position accuracy of the actuator can be improved in real time during continuous operation using a model based compensation method. The aim is to achieve a mean unidirectional position deviation of less than ±0.5μm while using only a single embedded sensor in the coil for temperature monitoring.
Model based compensation of thermal disturbance ina precision linear electromagnetic actuator
Jonathan Hey, Choon Meng Kiew, Guilin Yang, Ricardo Martinez-Botas
Abstract—Thermal disturbance is a major source of positioningerror in precision positioning systems. Conventional approachof using special materials for construction and sophisticatedgeometric design based on advanced computer simulation canbe costly as well as time consuming to implement. Moreover,dynamic thermal disturbances cannot be effectively compensatedfor by such methods. The approach presented in this paper usesmodel estimated position error coupled with sensor measurementas a feedback compensator of the output shaft position. A statespace model is used in a Modified Kalman Filter (MKF) to reducethe total number of temperature sensors needed for estimation ofthe thermally induced position error. A maximum temperatureestimation error of 1.8% of the measurement range is recorded.An online parameter estimation method is implemented to ‘finetune’ a transfer function model during a calibration stage beforecompensation. The model based compensation method resultedin a mean unidirectional positioning deviation of −0.2µm andrepeatability of ±0.7µm during a 5 hour continuous operation.
THERMAL disturbance is a major source of position-ing error in precision positioning systems. Positioning
error arises due to the internal or external thermal loadingwhich leads to thermal strain. Material deformation due totemperature changes causes deviation of the output positionfrom the desired position [1]. Such thermal error have beenminimized by using construction material with low coefficientof thermal expansion (CTE) and crafted geometric designs toensure the thermal center is at the output shaft [2]. Some havecooling system which actively maintains a constant operatingtemperature. However, such approaches can be costly, difficultto implement and not all the dynamic thermal error can beeliminated by these methods [1].
Fig. 1 shows a linear electromagnetic actuator designed forsubmicron positioning accuracy. Such actuators are commonin precision stages which require long range travel but theyare susceptible to disturbances [3]. However, the test deviceuses an electromagnetic driving scheme coupled with flexure-based supporting bearings to improve the position accuracy [4].Nevertheless, it still suffers from thermal disturbance due to the
Manuscript received xxxxx, 2013. This work is supported by the Agencyfor Science, Technology and Research (A*STAR) of Singapore.
Jonathan Hey and Ricardo Martinez-Botas are with the Mechanical En-gineering Department, Imperial College London, Exhibition Road, London,SW7 2AZ, UK (e-mail: [email protected], [email protected])
Choon Meng Kiew and Guilin Yang are with Mechatronics group, SingaporeInstitute of Manufacturing Technology, 71 Nanyang Drive, Singapore, 638075(e-mail: [email protected], [email protected])
internal heat generation at the coil during operation. Thermalexpansion of the output shaft leads to eventual positioningerror. A model based estimation of the disturbance coupledwith active output compensation is a cost effective methodto minimize the effects of dynamic disturbance [2],[5]. Therelationship between temperature increase and the resultingthermal disturbance is determined from experimental datathrough model identification techniques [6],[7].
Fig. 1: Flexure based electromagnetic linear actuator
Recent advances in this area of research focuses on system-atic reduction of temperature sensors required for modeling[8]. Parallel work on optimal temperature estimate of the rotorcoil in an induction machine is shown using Kalman filteringtechnique [9]. Strategies for fine tuning model estimation ac-curacy is reported in [7], [10] which shows the trend of onlineparameter estimation using iterative algorithms. In this paper,a model based compensation method is presented which showsits effectiveness in minimizing thermal disturbance in the testdevice. The compensation model is experimentally determinedfrom the measured input output data using the subspace andleast square method for model parameter estimation.
Internal temperature distribution is estimated using a statespace model coupled with measurement feedback from anembedded temperature sensor in the coil. A Modified KalmanFilter (MKF) is presented to show how sensor reduction canbe achieved in such an application. The thermo elastic processis modeled using this temperature information as input toa transfer function model. A recursive parameter estimationmethod is used to tune the transfer function model beforecompensation is carried out in an extended 5 hour test tosimulate continuous operations. The purpose of this work isto show how the position accuracy of the actuator can beimproved in real time during continuous operation using amodel based compensation method. The aim is to achieve amean unidirectional position deviation of less than ±0.5µmwhile using only a single embedded sensor in the coil fortemperature monitoring.
A model identification methodology is used to determinethe underlying mathematical models of the thermo elasticprocess. The method involves the selection of the appropriateexcitation signal, filtering of measured response, selection ofa suitable model structure, model parameter estimation andmodel validation. The selected model structures are the discretetime transfer function and state space model. The choice ofmodel structure and parameter estimation technique will befurther elaborated in the subsequent subsections.
A. Transfer function modelA discrete time Multiple Input Single Output (MISO)
transfer function model is used to model the position errorarising from temperature changes using the Output Error (OE)parameter estimation technique. The model is derived bycombining several Single Input Single Output (SISO) modelsin parallel. A general SISO model is presented mathematicallyin (1) where y is the output, u is the input while b,f are themodel parameters. n denotes the current time instance whilenb and nf are the number of past input and output data usedrespectively for estimating the current output.
y(n)+f1y(n− 1) + · · ·+ fny(n− nf )
= b1u(n− 1) + · · ·+ bnbu(n− nb)
(1)
A first order MISO model is chosen to model the thermalexpansion (yL) using discrete temperature points (uT ) asinputs. The MISO model can be expressed mathematically by(2) where m refers to the mth temperature point out of a totalof M points.
yL(n+ 1) = f1yL(n) +
M∑m=1
1∑r=0
bm,rumT (n− r) (2)
A total of 12 discrete temperature points are used as inputto the model as illustrated in Fig. 2 . Having more sensors andpast temperature measurements will enhance the estimationof the thermal expansion. But the first order MISO model isselected for its simplicity as required in a real time application.Nevertheless, it is possible to incorporate a longer time horizonwith such a model if extra computation power is available. Thesensor positions are selected to capture the changes in thermalgradients in the coil assembly and the surrounding structuredue to the outflow of heat from the coil over time.
Equation (2) can be expressed in a more compact form withmatrix notations as shown by (3) where ϕ and θ are the datavector and model parameter respectively.
Equation (4) defines the output error (e) as the differencebetween the measured output (y) and model output (y). Aleast square algorithm is used to determine θ that minimizes efor a set of data vector - {ϕ(1), · · · , ϕ(n), · · · , ϕ(N)} where
Fig. 2: Temperature points of interests on test device
N is the total number of sampled instances. The parameterestimation method is known as the Output Error (OE) methodsimply because it uses the output error as a measure of theresidual remaining from the parameter estimation process.
A block least square algorithm makes use of the whole set ofdata to estimate the model parameter in a single mathematicaloperation. The appended data matrix (Φ) is formed by stackingϕ row wise for each sampled instance. To ensure a goodestimate of the model parameters, the experiment durationshould be sufficiently long (in the order of hours) while thesampling rate is maximized (in order of ms) to reduce the lossof information and improve tracking accuracy. The dimensionof the data matrix is such that Φ ∈ R(2M+1)×(N). The solutionof a block least square problem requires the inversion of Φwhich is a costly mathematical operation.
A recursive algorithm is chosen because the solution iscalculated iteratively in small steps and uses an efficientmathematical substitution to eliminate the need for matrixinversion. The recursive least square algorithm is summarizedhere by (5) and it is is derived from the iterative solution ofthe block least square algorithm. The solution is proven to beequivalent to that of the block least square [11].
Q(n) =1
λ(n)
[Q(n− 1)− Q(n− 1)ϕT (n)ϕ(n)Q(n− 1)
λ(n) + ϕ(n)Q(n− 1)ϕT (n)
]L(n) =
Q(n− 1)ϕT (n)
λ(n) + ϕ(n)Q(n− 1)ϕT (n)
θ(n) = θ(n− 1) + L(n) [yL(n)− ϕ(n)θ(n− 1)](5)
It should be noted that an iterative solution would requirean initial estimate of the variables. Q is a measure of theconfidence in the estimated model parameter θ. A sufficientlylarge Q(0) should be chosen to move the initial estimate θ(0)towards its final solution [11]. λ that appears in (5) is a weightfactor for adjustment of the relative importance of the timedata used for model parameter estimation. However, such finetuning strategies are not exploited in the current work.
A recursive least square method have some practical ben-efits such as offering the flexibility of online model tuning
which is suitable for applications which require on-site setup calibration of the system. The user also has the flexibilityof choosing the calibration duration based on the maximumtime or level of residual error allowed before terminating theprocess. However, the method does require that computationbe completed within one sampling interval. This is possiblewith simple models like the first order MISO transfer functionmodel or when computation power is not a limitation.
B. State space modelTemperature changes in the system needs to be monitored to
estimate the thermal expansion as discussed in the precedingsection. The device is constructed with very high mechanicaltolerance to enhance electromagnetic performance. There islittle access to the internal components such as the internalmagnet assembly and bobbin (refer to Fig. 1). A state spacemodel is selected to model the dynamic temperature responseof the device at 12 points (refer to Fig. 2). Measurement froma single embedded sensor in the coil is used as feedback in aModified Kalman Filter for output adjustment. The details ofthe method will be presented in the next subsection.
A state space model is determined from the experimentallymeasured input-output data based on a combined stochastic-deterministic subspace method. A brief overview of the methodfrom an end users perspective is outlined in the next sectionfor completeness in presentation of the overall methodology. Itis not within the scope of this paper to discuss the formulationdetails which has been well documented in the pioneering work[12],[13]. The key steps outline in the next section are basedon work reported in [14].
It should be pointed out that the temperature used formodel identification refers to the temperature change fromthe ambient. Substituting the absolute temperature with thisrelative temperature will improve the modeling accuracy sinceambient temperature affects the overall heat loss of the device.The choice of input signal is important to ensure the properidentification of the model parameters based on the underlyingcausal relationship between the input and temperature rise.Eddy current, magnet hysteresis and mechanical losses whichare common to electromagnetic actuators are significant onlywhen there is large or high speed motion [15],[16].
For this test device the conversion losses come mainly fromthe resistive heating of the copper wires in the coil. Theresistive heating is derived from the electrical power suppliedwhich is given by the product of the supplied voltage (V ) andcurrent drawn (I) from the source and it is used as input tothe state space model. The choice of input waveform and itsfrequency content is also important for the proper identificationof the model which would be discussed in section V.
x(n+ 1) = Ax(n) + Bu(n)
+ K(n) [y(n)−Cx(n)−Du(n)]
y(n) = Cx(n) + Du(n) + [y(n)−Cx(n)−Du(n)]
(6)
A standard state space representation is shown in (6) wherey is the output and u is the input while A,B,C and D are thestate matrices. The innovation term, (y−Cx−Du), accountsfor the stochastic or noise component of the measured output
(y). K is the Kalman filter gain whose value is dependenton the noise content of the output. The state matrices andKalman gain are determined using the combined stochastic-deterministic subspace method.
1) Combined stochastic-deterministic state space modelidentification using the subspace method: The subspacemethod makes use of the geometric projection methods such asthe orthogonal and oblique projections of future and past input-output data sequence to extract the state matrices (A,B,C,D)and Kalman gain (K). The method is based on these funda-mental geometric operations and they are defined in (7)-(9)for completeness. The orthogonal projection of any matrix Yonto another matrix U is defined by (7). The projection ontoits orthogonal complement, denoted with a superscript ‘⊥’, isdefined by (8). The oblique projection of a matrix Y ontomatrix W along the row space of matrix U is given by (9)where the superscript ‘†’ denotes a pseudo inverse.
Y/U = YUT (UUT )−1U (7)
Y/U⊥
= Y[I−UT (UUT )−1U] (8)
YU/W = [Y/U⊥] · [W/U⊥]† ·W (9)
[Up
Uf
]=
u(0) · · · u(N − 1)u(1) · · · u(N)
......
...u(i− 1) · · · u(i+N − 2)u(i) · · · u(i+N − 1)
u(i+ 1) · · · u(i+N)...
......
u(2i− 1) · · · u(2i+N − 2)
(10)
The input and output data are organized into block matricesas shown by an example of the input block Hankel matrixin (10). i is the number of block rows of the data used foranalysis and it is user selectable but should be greater than thesystem order. It is composed of two parts which are denotedby the subscript ‘p’ and ‘f ’ to represent past and futuredata respectively. Past data includes data up to the instance(i−1) before while future data include data up to the instance(2i − 1) after as illustrated in (10). The column width of theblock matrix is determined by the total length of the datasequence denoted by N . A superscript of ‘+’ and ‘-’ representsa shift into the future or past respectively by one block row asillustrated in (11).
[U+p
U−f
]=
u(0) · · · u(N − 1)u(1) · · · u(N)
......
...u(i− 1) · · · u(i+N − 2)u(i) · · · u(i+N − 1)
u(i+ 1) · · · u(i+N)...
......
u(2i− 1) · · · u(2i+N − 2)
(11)
The input block matrices, Yf ,Yp,Y+p ,Y−f are defined in a
similar manner. A block Hankel matrix is a combination of
the input and output data sequence in one matrix. It also havethe same notations as the input and output block matrices asillustrated by the forward shifted block Hankel matrix in (12).
W+p =
[U+p
Y+p
](12)
The system matrices are arranged into block matrices. Theyare defined as the extended observability (Γi), lower block tri-angular Toeplitz (Hi) and the reversed extended controllability(∆i) matrix which are given by (13), (14) and (15) respectively.
Γi =
C
CA...
CAi−1
(13)
Hi =
D 0 · · · 0 0
CB D · · · 0 0...
.... . .
......
CAi−2 CAi−3B · · · CB D
(14)
∆i =[
Ai−1B Ai−2B · · · B]
(15)
With the reorganized data and system matrices, the input-state-output relations can be expressed in compact form givenby (16)-(17) where the superscript ‘d’ denotes the deterministicpart of the data while ‘s’ represents the stochastic component.Ai is the appended state matrix which appears in (17).
Yf = ΓiXdf + Hd
iUf + Ysf (16)
Xdf = AiX
dp + ∆d
iUp (17)
The identification process starts by taking the oblique pro-jection of the output block matrix onto the block Hankel matrixalong the row space of the input block matrix as definedin (18). The resulting expression, Oi, contains the extendedobservability matrix which is extracted using a Singular ValueDecomposition (SVD) of the weighted expression in (19). Thedecomposed data is represented in the standard notation whereS1 is the first n non zero singular value which serves as a guidefor selecting the model order. The selection of the weightingmatrices (W1,W2) is dependent on the algorithm selectedby the user. The relative merits of the different algorithm isdiscussed in detail in [10]. For the current application andpurpose, it is selected that W1 = W2 = I based on theN4SID algorithm [14]. The extended observability matrix isderived from the SVD process based on the definition givenby (20).
Oi = Yf/UfWp (18)
W1OiW2 = [ U1 U2 ]
[S1 00 0
] [VT
1
VT2
](19)
Γi = W−11 U1S
1/21 (20)
From the definition in (13), the state matrix (C) is givenby the top block row of the extended observability matrix(Γi). The system matrix (A) is related to it by the relationship
ΓiA = Γi where Γi and Γi is the extended observability ma-trix with the top and bottom block row removed respectively.Thus, the state matrix (A) is simply given by A = Γi
†Γi.The sequences Zi+1 and Zi are defined by a set of oblique
projections shown in (21). They are related to the Kalmanfiltered states Xi+1 and Xiby (22).
Zi+1 = Y−f /
[W+
p
U−f
], Zi = Yf/
[Wp
Uf
](21)
Zi+1 = ΓiXi+1 + Hdi−1U
−f , Zi = ΓiXi + Hd
iUf (22)
The stochastic component of the output is assumed to beuncorrelated to the Kalman filtered states. As such the stateequations first introduced in (6) can be written in block formgiven by (23) where Rw and Rv matrices are the stochasticcomponent of the output. Equation (24) is the state equationsexpressed in terms of Γi, A, C and K by using the definitions(22) to substitute for the states in (23). K is defined as theKalman gain sequence as shown in (25).[
Xi+1
Yi
]=
[A BC D
] [Xi
Ui
]+
[Rw
Rv
](23)
[Γ†i−1 · Zi+1
Yi
]=
[AC
]·Γ†i ·Zi+K·Uf +
[Rw
Rv
](24)
K =
[ (B|Γ†i−1 ·Hd
i−1
)−A · Γ†i ·Hd
i
(D|0)−C · Γ†i ·Hdi
](25)
Γi, A and C are already determined from the earlier analy-sis which leaves the Kalman gain sequence to be determined.K is obtained from a least square regression of (24). This ispossible since Rw and Rv are assumed to be stochastic noisesequence uncorrelated to Zi and Uf . Thus, Rw and Rv arethe residual errors from the least square regression process.
K1|1...
K1|iK2|1
...K2|i
= N
[DB
](26)
Equation (26) shows K as a linear combination of B andD where N is a function of the already known Γi, A andC. The final step to determine B and D uses the leastsquares regression on (26). The preceding discussion givesthe general description of the steps involved to determinethe state matrices A, B, C and D as first defined in (6)using the combined stochastic-deterministic subspace method.A solution is guaranteed since only linear algebraic operationssuch as the least square algorithm is used to solve the inversemathematical problem. Subspace algorithms are non-iterativewhich means there are no convergence issues. The method isrobust and efficient which makes it suitable for large data sets.
2) Modified Kalman Filter: In order to obtain an optimaltemperature estimate, sensor feedback as applied in a KalmanFilter (KF) is used to minimize the state space model output er-ror. The output error first presented in section II is reintroducedin (27) with a similar definition where y and y are the measuredand model output respectively. An additional term, y, is theadjusted output from applying the KF. Quantities derived fromthe adjusted output are termed posterior while those which arenot are termed priori and indicated by a superscript ‘-’. Theerror and error covariance are defined by (27)-(28).
e− = y(n)− y(n), e(n) = y(n)− y(n) (27)
P−(n)=E{[
e−(n)][
e−(n)]T}
,P(n)=E{[e(n)][e(n)]
T}
(28)
Temperature is estimated from the state space model duringthe ‘time’ update. During the ‘measurement’ update a weightedcorrection of the output is performed based on the measuredoutput error and the Kalman gain (K) as illustrated in Fig. 3.
Fig. 3: Kalman Filter (KF) applied to a state space model
‘Time update’
x(n + 1) = Ax(n) + Bu(n)
P−(n+ 1) = AP(n)AT + Rw(29)
‘Measurement update’
K(n+ 1) = P−(n+ 1)CT[CP−(n+ 1)CT + Rv
]−1P(n+ 1) = P−(n+ 1)−K(n+ 1)CP−(n+ 1)
x(n+ 1) = x(n+ 1)
+ K(n+ 1) [y(n+ 1)−Cx(n+ 1)](30)
The mathematical operations during the ‘time’ and ‘mea-surement’ update are shown in (29)-(30) [9]. The amount ofoutput correction is determined by the Kalman gain (K). Thevalue of K is derived iteratively based on a recursive leastsquare algorithm which minimizes P over time. Rw, is addedto P as shown in (29) at each iteration. It is represents themodel output error during the ‘time’ update. Similarly, Rv , isadded to P during the ‘measurement’ update which reflectsthe amount of measurement uncertainty. K and Rv have aninverse relationship as shown by (30). Thus, output correctionwill be less when the measurement uncertainty is high. K, Rw
and Rv was first introduced in the preceding section. The useof the same notation indicates that they are the same variable.The difference lies in way it is calculated; direct or iterative.
The state space model is represented in (31)-(33) withaltered notations to illustrate the physical quantities it rep-resents. yT is the model estimated temperature while uq isthe input heat generation to the system and A, B, C, Dare the state matrices defined in (6). D is set to be a null
matrix during the identification process since the output is notdirectly correlated with the input. Equation (32) simplifies togive x = C−1yT which directly relates the internal states tothe output. Substitution of this relation into (31) leads to thetransformed state equations (33).
x(n+ 1) = Ax(n) + Buq(n) (31)
yT (n) = Cx(n) + Duq(n) (32)
y(n+ 1) = CAC−1yT (n) + CBuq(n) (33)
The KF is presented earlier to highlight the changes madein the Modified Kalman Filter (MKF) as illustrated by Fig. 4and represented by (34)-(35) with altered notation for differ-entiation. The modification of the KF is aimed at reducing thenumber of sensor measurement needed for output correction.The following is a summary of the modifications made.
1) Replace the state (x) with the output (yT ) by substitu-tion of the original state matrices (A, B, C) with thetransformed state matrices (α = CAC−1, β = CB)
2) Output correction using an output mapping (ζσ) of themeasured error signal
Fig. 4: Modified Kalman Filter(MKF)
‘Time update’
yT (n+ 1) = αyT (n) + βuq(n)
P−(n+ 1) = αP (n)αT +Rw(34)
‘Measurement update’
K(n+ 1) = P−(n+ 1)[P−(n+ 1) +Rv
]−1P (n+ 1) = P−(n+ 1) [1−K(n)]
yT (n+ 1) = yT (n+ 1)
+K(n+ 1) [yT,1(n+ 1)− yT,1(n+ 1)] ζσ
(35)
The output mapping (ζσ) shown in (35) is derived from thecovariance between the error signal measured at coil to theother temperature points as defined in (36). The covariancefor sequences of data (z1, · · · , zi, · · · , zM ) can be calculatedusing (37) where µ is the mean of the each data sequence.The output error sequence is obtained from the results of themodel identification test which will be presented in section V.
ζσ =[
1 · · · σj
σ1· · · σM
σ1
]T(36)
σj =1
N
N∑i=1
[z1(i)− µ1] [zj(i)− µj ] (37)
In the original KF, the state matrix (A) in (29) serves asa weighting factor for error covariance updating. In the MKF,the Kalman gain (K) and error covariance (P ) are calculatedbased on a single measured error signal. Equation (34) reflects
the reduced dimension in which α is obtained from thefirst row of the transformed state matrix. In the MKF, Rv
corresponds to the thermocouple measurement uncertainty. Rw
is estimated from the measurement uncertainty and the residualerror resulting from the subspace identification process. Themeasurement uncertainty and residual error will be presentedin section IV and V respectively.
Subsequent application of the filter would require only themeasurement feedback from the embedded temperature sensorat the coil. Application of the MKF will reduce the total num-ber of sensors needed for estimation of the thermal expansion.The assumption is that unmeasured error signals are correlatedbased on a statistical measure, given by the covariance ofthe output error predetermined during a calibration test beforethe actual filter application. The improvement in estimationaccuracy will be discussed in section V.
III. COMPENSATION MODEL AND POSITION CONTROLLER
The compensation model is used as an estimate of theposition error based on the measured heat generation (uq) andcoil temperature (yT,1). The model can be described entirelyby (38)-(39). yT refers to the adjusted temperature estimateafter Kalman filtering. The estimated thermal expansion (yL)is simply a product of the adjusted temperature estimate againstthe model parameter (θ) shown in (39). The model is illustratedin a block diagram in Fig. 5.
The actuator can be represented by a series combination ofan electrical and mechanical subsystem as illustrated in Fig.6. The electrical subsystem is a series connection of a resistorand inductor where R and L are the resistance and inductancerespectively. The mechanics can be modeled as a mass in serieswith springs and a damper. The moving mass (m) is made upof the coil and bobbin mass while the spring force is providedby the flexure mechanism with a spring constant k. Dampingis caused by eddy current induced in the bobbin-coil assemblywith a damping coefficient of b. Ka is the amplifier gain while
kT is the force constant which is dependent on strenght of thecoupling magnetic field and the number of turns of the coil.
Fig. 6: Actuator electromechanical model
The electro-mechanical and thermo elastic interaction in thedevice is illustrated in Fig. 7 by the system block diagram.The uncompensated position is measured by an optical positionencoder. The compensation model estimated thermal expansionis combined with the position measurement to give the com-pensated position. Positioning of the output shaft is controlledwith a PID controller using the compensated position as afeedback. A second position encoder is used to measure thereference output position for performance characterization.The position error is defined as the difference between thisreference output position and the target position.
Fig. 7: System block diagram with feedback compensation
All the quantities indicated in the block diagram are mea-sured or derived from measurements except for the com-pensated position which is obtained from model estimation.Since the dynamic response of the actuator is measured, anempirical PID controller tuning method is used to determinethe controller gains. The method is based on the Ziegler-Nichols standard tuning approach by analyzing the open loopstep response of the system. The continuous form of a standardPID controller is given by (40) where ε is defined as thefeedback error signal and υ is the output corrective action.Kp is defined as the proportional gain while Ti and Td are theintegral and derivative time respectively.
D(s) =υ(s)
ε(s)= Kp
(1 +
1
Tis+ Tds
)(40)
Fig. 8 shows the open loop step response of the system. Thefirst step towards a tuned controller starts with analyzing theopen loop step response of the system. The three parametersobtained from the output response are termed the delay time(τd), time constant (τc) and output gain (K0) as illustrated inFig. 8. The controller gains are estimated from these threeparameters based on the relationship presented in Table I.A discrete version of the PID controller is necessary forimplementation on the actual system. Equation (41) describesthe general discrete form where, TS = 1ms, is the samplingtime of the system [17].
TABLE I: Controller parametersProportional gain Integral time Derivative timeKp = 1.2τc/(K0τd) Ti = 2τd Td = 0.5τd
Inital estimate 0.021 0.004 0.001
υ(n) = υ(n−1)+
KP
[(1+
TsTi
+TdTs
)ε(n)−
(1+
2TdTs
)ε(n−1)+
TdTsε(n−2)
](41)
The controller gains are adjusted based on the initial es-timates using the following guidelines; (i) proportional gainis used to decrease the rise time (ii) integral gain eliminatessteady state error and (iii) derivative gain reduces overshootand settling time. The close loop system incorporated withthe compensation model has altered dynamics. Further tuningwould be necessary, until a satisfactory dynamic performanceis achieved. A speed limiter sets the maximum actuation speedto 1mm/min to prevent large step changes and instability.
IV. MEASUREMENT AND EXPERIMENTAL TEST CASES
Fig. 9: Experimental set up with (a) position encoders &thermocouples (b) air temperature measurement and (c) side view
The experimental set up for device testing is shown togetherwith the measurement system in Fig. 9. The optical linearposition encoders and K type thermocouples can be seen inFig. 9(a). The air temperature is measured at the point shownin Fig. 9(b) and (c) shows the side view of the device. Themeasurement uncertainties associated with the position andtemperature measurement are summarized in Table II.
TABLE II: Measurement accuracy and resolutionInstrument Measured
quantityAccuracy Resolution
MicroE Mercury 3500optical linear encoder
Position ±1µm (long range)±0.15µm (short range)
±5nm
K type thermocouple Temperature ±1.0◦C ±0.25◦C
The optical encoders have a built in digital interpolatorwhich gives position measurement of up to 5nm in resolu-tion. However, the uncertainty of the measurement typicallyamounts to 1% of grating size (20µm) of the scale for shortrange measurement. Moving average filters are used to removesome of the unwanted noise due to digitization. A total of fourtests were conducted for model identification and validation.The control parameter, input signal, measured and derivedquantities together with the objective of each test are shownin Table III.TABLE III: Tests conducted for model identification and validation
No Inputsignal
Derivedquantity
Measuredquantity
Objective Controlparameter
1 Step Electricalpower
Voltage,Current,Temperature
Thermal timeconstantdetermination
Current
2 Varyingpulsewidth
Electricalpower
Voltage,Current,Temperature
State spacemodelidentification,Kalman filterparameterdetermination
Current
3 Varyingperiodtrapezoidal
Electricalpower
Voltage,Current,Temperature,Position
State spacemodelvalidation
Position
4A Varyingperiodtrapezoidal
Electricalpower,Positionerror
Voltage,Current,Temperature,Position
Transferfunctionmodelidentification
Position
4B Fixedperiodtrape-zoidal
Electricalpower,Positionerror
Voltage,Current,Temperature,Position
Compensationmodelvalidation
Position
V. RESULTS AND DISCUSSION
Test 1 is a step input of electrical power (3.2W) to obtainthe temperature response and shortest thermal time constantof the system. This step response analysis serves as a basisfor selecting the input signal for model identification in test 2.Fig. 10 shows the coil and bracket temperature which are thenearest and furthest points from the heat source respectively.
Fig. 10: Test 1 - Thermal time constant determination
The temperature response resembles a first order systemwhich is characterized by an initial gradient of 1/τc and azero gradient approaching the steady state output. τc is thetime constant defined as the time taken for the output to reach63.2% of the final steady state value as illustrated in Fig. 10.The system response will be attenuated beyond the cut offfrequency which is defined as fc = 1/2πτc.
Therefore, to enhance the Signal to Noise Ratio (SNR) ofthe measured output for the thermal model identification intest 2, the input waveform is designed based on a frequencydomain analysis. A varying pulse width input signal is chosenbecause of its flexibility in design. The frequency content ofthe signal is controlled by changing the duration of the pulses.A total of 3 pulses of varying period is used for input excitationduring the ‘heat up’ period followed by a period of zero inputto allow the system to ‘cool down’ as shown in Fig. 11(a).The device undergo a similar ‘cool down’ period during actualoperations. The heat loss at the natural convective boundaryundergo changes between these two periods. Thus, it wouldlead to a more robust model if they are accounted for duringthe identification test.
Fig. 11: Test 2 - State space model identification: input waveform in(a) time and (b) frequency domain
The input power spectrum is expected to be low due to thelong period of zero input. It is critical to engineer the inputsignal to enhance its power content in the desired frequencyband. The shortest time constant recorded, τc = 370s, corre-sponds to a cut off frequency of fc = 4.3 × 10−4Hz whichgives an upper limit estimate of the device thermal responsebandwidth. The selected waveform has an increasing powerspectrum in the desired frequency band as shown in Fig. 11(b).This input signal is used for thermal model identifcation in test2. Fig. 12(a)-(b) shows the measured temperature response andcorresponding output error plot is shown in Fig. 12(d)-(f). Thiserror sequence is used to determine the Modified Kalman Filter(MKF) parameter.
〈e〉 = [ e1 · · · em · · · e12 ]T
where em =
√∑Nn=1 [yT,M (n)− yT,M (n)]
2
N
(42)
〈e〉 is a measure of the output error averaged over thewhole measurement period as defined by (42). 〈e〉 is termedthe residual error in this instance since it is calculated fromthe output error of this identification test. The estimated coiltemperature has a residual error of 0.46◦C which is usedfor estimating the filter parameter, Rw, as defined in (34).
The output mapping defined as ζσ =[1 · · · σj
σ1· · · σ12
σ1
]Tis
calculated using (37) and tabulated in Table IV.TABLE IV: Output mapping (ζσ)
Fig. 12: Test 2 - State space model identification: (a-c) temperatureresponse and (d-f) output error
Test 3 is designed for the characterization of the temperatureestimation accuracy with and without Kalman filtering. Theinput motion profile is designed to induce significant thermalloading on the device. It is separated into the ‘heat up’ and‘cool down’ period as shown Fig. 13(a). The first 5 cycle(s)of the trapezoidal motion path during the ‘heat up’ period isshown in Fig. 13(b). This path simulates a repeated maximumforward and backward stroke with varying duration of holdin between as illustrated in Fig. 13(c). This ensures that thedevice is put through the typical range of motion expectedduring the actual operation.
There is significant temperature rise recorded at all thepoints of interest as shown in Fig. 14(a)-(c) due to this inputmotion profile. The largest temperature rise is recorded at
Fig. 13: Test 3 - State space model validation: (a) input motionprofile, (b) 5 cycle(s) and (c) 1 cycle of the varying period
trapezoidal motion path
the coil with a final temperature of 27.9◦C after an hour ofoperation. There is a large temperature gradient between coiland outer stator as shown in Fig. 14(a). This indicates a highresistance to heat flow from the internal components outwards.One of the major barrier to heat transfer is the internal air gapbetween the coil and magnet assembly (refer to Fig. 1). Theresulting temperature rise represents an accumulation of heatwithin the device which is undesirable as it gives rise to theeventual thermal expansion and positioning error.
A comparison of the model output is made against themeasurement where the output error plot is shown in Fig.14(d)-(f). A maximum output error of 0.56◦C is recorded atthe inner magnet as shown in Fig. 14(e) with a time averagederror of 0.26◦C. From the error plots, it is apparent that themodel estimation accuracy is affected by the device operatingcondition. The ‘heat up’ and ‘cool down’ period registers verydifferent error signature. During the ‘heat up’ period the modelis overestimating the temperature while the opposite is trueduring the ‘cool down’ period. There is also a slow drift in thetemperature estimate away from the measurement during the‘cool down’ period which is caused by the changing conditionswithin the test chamber. Some of these modeling inadequaciescan be overcome with model output adjustment using sensorfeedback in the Modified Kalman Filter (MKF).
Fig. 15 shows the output error after applying the MKF.The maximum error originally recorded at the inner magnetis now reduced to 0.16◦C with an average of 0.05◦C overthe measurement period. With application of the MKF, themaximum error recorded across the 12 temperature points ofinterest amount to 0.37◦C which is equivalent to 1.8% of themeasurement range. The maximum and time averaged outputerror for other temperature points are tabulated in Table Vand Table VI respectively. The error remaining after Kalmanfiltering is shown in brackets in both tables. As expected, thelargest reduction in error is observed for the coil temperature.The model validation result show that the MKF is able correctthe model output towards the actual temperature based onoutput mapping of the measured error signal.
Fig. 14: Test 3 - State space model validation: (a-c) temperatureresponse and (d-f) output error before filtering
TABLE V: Maximum output error (with) and without filteringCoil Bobbin Inner
Fig. 15: Test 3 - State space model validation: output error afterfiltering
Fig. 16: (a) Kalman gain and (b) error signal
Fig. 16(a) shows the Kalman gain reaching a steady valuebeyond the first 5 samples of data as the model outputapproaches the measured value. This means a decreasingerror signal over time. The mapped error signal is reducedproportionately due their linear relationship as shown by a oneof the selected signal in Fig. 16(b). Output correction will bereduced over time until a large error is induced possibly bysome external disturbance.
Output mapping provides an indirect estimate of the outputvariation at the unmeasured temperature point based on themeasured error signal and the statistical characterization of thesystem in a prior test. However, if the cause for temperaturevariation is external, such as an additional heat source, it willfor example have more effect on the outer stator temperaturethan the coil temperature. This disturbance will not be ad-equately captured by the embedded sensor. So long as thesystem is not subjected to significant changes, the MKF isan effective tool to reduce the short term tracking and longterm drift error in the thermal model output. At this point, thethermal model is fully calibrated from the previous three tests.The remaining temperature sensors are removed except the
embedded sensor at the coil. Temperature is estimated usingthe state space model coupled with the MKF in this final test.
Fig. 17: Test 4 - Extended test: (a) input motion profile, (b) 5cycle(s) and (c) 1 cycle of the fixed period trapezoidal motion path
Test 4 is divided into two parts (4A and 4B) with a total offour segments as illustrated in Fig. 17. Test 4A is a single twohour segment which consists of a continuous cycle of varyingperiod trapezoidal motion path for one hour followed by onehour of cool down. The varying period trapezoidal motion pathis similar in design to the one used in test 3 (refer to Fig.13). This test is designed for the identification of the transferfunction model. The model output is taken as the measuredposition error defined in section III. The temperature responseresulting from this motion profile is shown in Fig. 18(a)-(c).There is significant temperature variation due to the cyclicalthermal loading which is ideal as the input for fine tuning thetransfer function model parameters.
Test 4B has 3 one hour segments which each consist of acontinuous 32 cycle(s) of 75s fixed period trapezoidal motionpath followed by a ‘cool down’ period as shown in Fig. 17(a).The fixed period trapezoidal motion path is shown in Fig.17(b)-(c). This test is used for validation of the compensationmodel. This validation test is designed to simulate a continuousoperation. The device will undergo cycles of thermal load-ing during the ‘heat up’ period and significant temperaturevariation is observed as a result of this motion profile. Themaximum temperature recorded during this extended test is27.6◦C as shown in Fig. 18(a). The physical thermal boundaryconditions such as the chamber air temperature is expectedto change over the course of the test. This would in turnaffect the compensation model estimation accuracy. However,the effect of changing boundary conditions can be minimizedwith minimal temperature sensing using the Modified KalmamFilter. This would help improve the positioning error trackingability of the compensation model.
The positioning error before compensation is shown inFig. 18(d). The mean unidirectional position deviation beforecompensation is +5.7µm with a repeatability of ±7.3µm forthe duration of the extended test (calculated based on ISO230-2). A SISO transfer function model which uses only the
Fig. 18: Test 4 - Extended test: (a-c) temperature estimate with Modified Kalman Filtering, (d) position error and (e) position error residualafter compensation
measured coil temperature as an input is used as the estimatorof position error for comparison with the compensation model.The model estimated position error from both models areplotted against the measurement in Fig. 18(d) and the residualsare shown in Fig. 18(e). The residual is minimized during thefirst test segment as the position error is measured and used formodel calibration. Thereafter, the measurement is used solelyfor validation as shown in test segments 2-4 of Fig. 18(e).
The SISO transfer function model is able to reduce theposition error down to a mean deviation of +0.7µm with a re-peatability of ±2.3µm. With implementation of the compensa-tion model in a feedback arrangement, the mean unidirectionalposition deviation have been reduced even further to −0.2µmwith a repeatability of ±0.7µm. The positioning accuracyachieved with the proposed method shows a threefold increaseover the expected improvement if a SISO transfer function
model is implemented instead. The model based compensationmethod is effective in reducing a large component of theposition error caused by thermal loading. It shows good shortterm tracking ability as well as long term stability.
VI. CONCLUSION
This work illustrates the effect of dynamic thermal distur-bance on positioning accuracy of a precision linear electromag-netic actuator. A model based compensation method is appliedto the test device. It resulted in a final mean unidirectionalposition deviation of −0.2µm with a repeatability of ±0.7µmduring a continuous 5 hour test. The significant improvementin positioning accuracy is due to the robust estimation of theposition error using the compensation model. The model isdetermined from an identification procedure. Online modeltuning is applied in an extended test to illustrate its practical
usefulness. Output mapping of the measured error signal fromthe embedded temperature sensor in a Modified Kalman Filter(MKF) resulted in minimal temperature sensors required foractive position compensation. A total of only 1 sensor isneeded from an original 12. Application of a MKF resulted inimproved temperature estimates with absolute error of 0.37◦Cwhich is equivalent to 1.8% of the measurement range.
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Jonathan Hey received the B.Eng (Hons) Degreein Mechanical Engineering from Nanyang Techno-logical University (NTU), Singapore, in 2008. Heis a recipient of the A*STAR graduate scholarship.He is currently a candidate for the Ph.D. degree atImperial College London, UK. His research inter-est is on thermal management of electromechanicaldevices focused on disturbance model identification,compensation methods and thermal design analysis.
Choon Meng Kiew received the B.Eng (Hons)Degree in Electrical and Computer Engineering fromNational University of Singapore (NUS) in 2003.After which, he was awarded A*STAR GraduateScholarship to pursue his Ph.D. studies at NUS andwas conferred the degree in 2008. He joined Sin-gapore Institute of Manufacturing Technology sinceSept 2007. His research interests are in the area ofthermal process control, precision control of nano-positioning stages, control of web handling systemand also thermal related analysis
Guilin Yang received the B. Eng degree and M.Eng degree from Jilin University, China, in 1985 and1988 respectively, and Ph.D. degree from NanyangTechnological University in 1999. From 1988 to1995, he had been with the School of Mechanical En-gineering, Shijiazhuang Tiedao University, China, asa lecturer, a division head, and then the vice dean ofthe school. Since 1998, he has been with SingaporeInstitute of Manufacturing Technology (SIMTech),Singapore. Currently, he is a senior scientist and themanager of the Mechatronics Group. His research
interests include precision mechanisms, electromagnetic actuators, parallel-kinematics machines, modular robots, industrial robots, and rehabilitationdevices. He has published over 190 technical papers in referred journals andconference proceedings, 8 book chapters, and 2 books. He has also filed12 patents. He was the committee chair of Singapore IEEE Robotics andAutomation Chapter (2011- 2012) and the Technical Editor of IEEE/ASMETransaction on Mechatronics (2008-2012). He is now the Editorial BoardMembers of Frontiers of Mechanical Engineering and International Journal ofMechanisms and Robotic Systems, and an associate editor of IEEE Access.
Ricardo Martinez-Botas is a Professor of Turbo-machinery at Imperial College London. He has anMEng (Hons) Degree in Aeronautical Engineeringfrom Imperial College London and a doctoral degreefrom the University of Oxford. He has developedthe area of unsteady flow aerodynamics of small tur-bines, with particular application to the turbochargerindustry. The contributions to this area centre on theapplication of unsteady fluid mechanics, instrumen-tation development and computational methods. Hehas expanded his research in the area heat transfer
of electrical machines and batteries. He has published extensively in journalsand peer reviewed conferences. He is currently Associate Editor of the Journalof Turbomachinery and is a member of the editorial board of two otherinternational journals. He is currently the Theme Leader for Hybrid andElectric Vehicles of the Energy Futures Lab at Imperial College.
